The lorenz attractor and a related population model

Parry, William

doi:10.1007/bfb0063293

Cited by 29 publications

(37 citation statements)

References 8 publications

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“…On that set, the map is smoothly conjugated to a piecewise affine Lorenz-type map of the interval. Results on the dynamics of Lorenz maps [33,41] immediately imply the following conclusion.…”

Section: Properties Of the One-dimensional Map G 1supporting

confidence: 56%

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Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

Fernandez

2014

J Stat Phys

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To identify and to explain coupling-induced phase transitions in Coupled Map Lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive.In this paper, we consider a family of CML composed by piecewise expanding individual map, global interaction and finite number N of sites, in the weak coupling regime where the CML is uniformly expanding. We show, mathematically for N = 3 and numerically for N 3, that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics.Despite that it only addresses finite-dimensional systems, to some extend, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension.August 28, 2013.

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Section: Properties Of the One-dimensional Map G 1supporting

confidence: 56%

“…The properties of the dynamics of G ,1 , in particular transitivity and ergodicity everywhere in the expanding domain (Proposition 4.1 in section 4.1), are then immediate consequences of the analogous results for symmetric piecewise affine Lorenz maps, see [33] for a detailed description. Letting…”

Section: A Analysis Of the Dynamics Of G 1mentioning

confidence: 80%

Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

Fernandez

2014

J Stat Phys

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“…Since the pioneering work of Rényi [36] and Parry [31,32,33,34], an increasing amount of attention has been paid to maps of the unit interval. Their study has provided solutions to practical problems within biology, engineering, information theory and physics.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Continuity of entropy for Lorenz maps

Zoe

Pearse

Quackenbush

et al. 2020

Indagationes Mathematicae

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“…Since the pioneering work of Rényi [29] and Parry [26,27,28], intermediate β-shifts have been well-studied by many authors, and have been shown to have important connections with ergodic theory, fractal geometry and number theory. We refer the reader to [18,30] for survey articles concerning this topic.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%